International Conference on Control, Games and Stochastic Analysis
October 29 – November 01, 2018
Hammamet, Tunisia
http://pinguim.uma.pt/Investigacao/Ccm/icsaa18/
University of Tunis El Manar, Faculty of Sciences of Tunis Ecole Polytechnique, France Université Paris Dauphine University of Oslo, Norway ETH Zürich, Switzerland
Hôtel Golden Tulip Zone Touristique Yasmine Hammamet Hammamet - 8050, Tunisie
1 Programme overview
Monday Tuesday Wednesday Thursday
8:50 Ekeland Streit, Agram Cardaliaguet Aïd 9:20 Beiglböck Blanchard, Obata Hamadène Bahlali 9:50 Frittelli Øksendal Fuhrman Zervos
10:50 Pontier von Waldenfels Hillairet Essaky 11:20 Horvath, Prömel Di Nunno, Hilbert Tan Draouil, Ayed Ben Salem, Burzoni Rüdiger, Silva Hu, Sgarra Friesen, Salhi Benezet Cipriano Kazi-Tani Kremer
16:00 Hubert, Hernández Scotti, M’rad Petterson Martin Maggis Cayé, Ben Alaya 17:00 Obłój Tankov Ouknine 17:30 Alfonsi Mezerdi Gozzi 18:00 Cox Kupper El Karoui
rien du tout Infinite dimensional Analysis and White Noise In memory of Takeyuki Hida
2 Monday, October 29
Chair: Dylan Possamaï
08:30 – 08:50 Opening 08:50 – 09:20 Ivar Ekeland Some thoughts on speculation 09:20 – 09:50 Mathias Beiglböck Causal transport and its role in mathematical finance 09:50 – 10:20 Marco Frittelli On fairness of systemic risk measures
= Coffee Break =
Chair: Mathias Beiglböck
10:50 - 11:20 Monique Pontier PDE for joint law of the pair of a continuous diffusion and its running maximum 11:20 – 11:40 Blanka Horvath Learning rough volatility 11:40 – 12:00 David Prömel On Skorokhod embeddings and Poisson equations 12:00 - 12:20 Sarah Ben Salem An equilibrium model of prevention effort under distortion risk measures 12:20 – 12:40 Matteo Burzoni Risk measures based on benchmark loss distributions 12:40 – 13:00 Cyril Benezet Partial hedging: a numerical scheme
= Lunch =
Chair: Mihail Zervos
16:00 – 16:20 Emma Hubert Optimal remuneration of correlated consumers in elec- tricity demand 16:20 – 16:40 Nicolás Hernández Santibáñez Contract theory in a VUCA world 16:40 – 17:00 Jessica Martin A revisit of the Borch rule for the principal–agent risk– sharing problem
Chair: Monique Jeanblanc
17:00 – 17:30 Jan Obłój On numerical techniques for robust methods of pricing and hedging 17:30 – 18:00 Aurélien Alfonsi Sampling of probability measures in the convex order by Wasserstein projection 18:00 – 18:30 Alexander Cox Utility maximisation with model–independent trading constraints
= 20:00 Dinner =
3 Tuesday, October 30
Infinite dimensional Analysis and White Noise In memory of Takeyuki Hida
Chair: Habib Ouerdiane
08:50 – 09:10 Philippe Blanchard ETH–approach to quantum mechanics 09:10 – 09:30 Ludwig Streit Fractional Brownian loops and trees 09:30 – 09:50 Barbara Rüdiger The Enskog equation and the Enskog–Boltzmann process 09:50 – 10:10 Nacira Agram Forward/backward stochastic Volterra integral equations and related topics 10:10 – 10:30 Bernt Øksendal Optimal control of SPDEs with space–mean dynamics
= Coffee Break =
Chair: Barbara Rüdiger
11:00 – 11:20 Nobuaki Obata Component sizes in random graphs 11:20 – 11:40 Wilhem von Waldenfels The H−Theorem in radiation transfer 11:40 – 12:00 Astrid Hilbert A mean–field game of pure jump type with common noise 12:00 – 12:20 Fernanda Cipriano On the optimal feedback controls for stochastic second grade fluids 12:20 – 12:40 José Luis Silva The Stein characterization of M−Wright distributions 12:40 – 13:00 Giulia Di Nunno Integration with respect to Lévy driven Volterra processes
======Excursion and lunch: the winery tour ======
======20:00: Dinner ======
4 Wednesday, October 31
Chair: Bruno Bouchard
08:50 – 09:20 Pierre Cardaliaguet Mean field games with a major player 09:20 – 09:50 Saïd Hamadène Mean–field backward–forward stochastic differential equations and mean–field non–zero sum stochastic differential games 09:50 – 10:20 Marco Fuhrman Optimal switching problems with an infinite set of modes: an ap- proach by randomization and constrained BSDEs
= Coffee Break =
Chair: Giulia Di Nunno
10:50 - 11:20 Caroline Hillairet Construction of an aggregate consistent utility 11:20 - 11:50 Xiaolu Tan On the planification problem in Mean Field Games 11:50 - 12:10 Kaitong Hu Continuous-time Principal-Agent Problem in a Linear Partially Ob- served System 12:10 - 12:30 Carlo Sgarra A forward price model for power markets based on branching pro- cesses 12:30 - 12:50 Nabil Kazi-Tani An open problem in ruin theory and its diffusion approximation regime
= Lunch =
Chair: Caroline Hillairet
16:00 - 16:20 Simone Scotti An optimal dividend control problem with investment opportunities and business cycle 16:20 - 16:40 Mohamed M’rad Markovian dynamic utility from monotonic characteristic 16:40 - 17:00 Marco Maggis Stochastic dynamic utilities and inter–temporal preferences
Chair: Bernt Øksendal
17:00 – 17:30 Peter Tankov Mean field games of optimal stopping: a relaxed control approach 17:30 – 18:00 Brahim Mezerdi On stochastic control of mean–field systems 18:00 – 18:30 Michael Kupper Computation of optimal transport and robust risk aggregation with neu- ral networks
= 20:00 Conference Dinner =
5 Thursday, November 1
Chair: Youssef Ouknine
08:50 – 09:20 René Aïd Optimal electricity demand response contracting 09:20 – 09:50 Khaled Bahlali Unbounded BSDEs driven by z2/y and application 09:50 – 10:20 Mihail Zervos Self–enforcing risk-sharing arrangements in a continuous–time endowment economy
= Coffee Break =
Chair: Pierre Cardaliaguet
10:50 – 11:20 El Hassan Essaky Generalized doubly reflected BSDE with predictable obstacles and stochastic quadratic growth 11:20 – 11:40 Olfa Draouil Optimal stopping of Hunt processes 11:40 – 12:00 Wided Ayed An extension of the It¯o integral 12:00 – 12:20 Martin Friesen On multi–type branching processes with immigration 12:20 – 12:40 Naoufel Salhi Some facts about Gaussian integrators 12:40 – 13:00 Jonas Kremer Existence of limiting distribution for affine processes
Chair: Jan Obłój
16:00 – 16:20 Roger Petterson On an epidemic diffusion model 16:20 – 16:40 Thomas Cayé Utility maximization in a multidimensional market with small nonlinear price impact 16:40 – 17:00 Mohamed Ben Alaya Local asymptotic properties for Cox–Ingersoll–Ross process with dis- crete observations
Chair: Nizar Touzi
17:00 – 17:30 Youssef Ouknine Non linear optimal stopping problem and Reflected BSDE in the pre- dictable setting 17:30 – 18:00 Fausto Gozzi Optimal control of SPDEs and path dependent SDEs: some recent results 18:00 – 18:30 Nicole El Karoui Stochastic utility function and optimization: a new point of view on market equilibrium
= 20:00: Dinner =
6 Titles and abstracts AGRAM Nacira (University of Biskra) Forward/backward stochastic Volterra integral equations and related topics
Abstract: Stochastic Volterra integral equations are a special type of integral equations. They represent inter- esting models for stochastic dynamics with memory, with applications to e.g.engineering, biology and finance. Solutions of stochastic Volterra (integral) equations are not Markov processes, and therefore classical methods, like dynamic programming, cannot be used to study optimal control problems for such equations. However, we show that, by using Hida-Malliavin calculus, it is possible to formulate a modified functional type of maximum principle suitable for such systems. An adjoint process obtained will be a backward stochastic Volterra integral equations with jumps. An interesting motivation of this type of equation is recursive utility and dynamic risk measures. Since backward stochastic Volterra integral equations are not in general a semimartingale, we will discuss some cases. This talk is based on joint works with Bernt Oksendal (University of Oslo, Norway) and Samia Yakhlef (University of Biskra, Algeria).
AÏD René (Université Paris Dauphine) Optimal electricity demand response contracting with responsiveness incentive
Abstract: We address the moral hazard underlying demand response contracts in electricity markets, we for- mulate the interaction problem between a producer and a consumer by means of a Principal-Agent problem. The producer, acting as the Principal, is subject to the generation costs related to the level and the volatility of generation, thus accounting for the limited flexibility of production. Based on the continuous-time consumption of the consumer, acting as the Agent, she sends an incentive compensation in order to encourage him to reduce his average consumption and to improve his responsiveness defined as the volatility of his consumption. We provide closed–form expression for the optimal contract that maximizes the utility of the principal in the case of linear energy valuation. We provide rationality for the form of the observed demand response contracts, that is a fixed premium for enrolment and a proportional price for the energy consumed. However, we show that the premium should be a function of the duration of the demand response event and that it should be negative in case of events of long duration. Further, we show that when the risk is optimaly shared between the two actors by the means of the optimal contract, the electric system can bear more risk as the resulting consumption volatility may increase. We calibrate our model to publicly available data of the Low Carbon London Pricing Trial, and we infer that a significant increase of responsiveness can be expected by the implementation of the control of the consumption volatility. We find that the responsiveness control would lead a significant increase of the value of the producer. We examine the stability of our explicit optimal contract by performing appropriate sensitivity analysis, and show that the approximation consisting in the linear energy valuation provides a robust approximation of the optimal contract. Joint work with Dylan Possamaï and Nizar Touzi.
ALFONSI Aurélien (École des Ponts ParisTech) Sampling of probability measures in the convex order by Wasserstein projection
Abstract: For µ and ν two probability measures on Rd with finite moments of order ρ ≥ 1, we define the respective projections for the Wρ-Wasserstein distance of µ and ν on the sets of probability measures dominated by ν and of probability measures larger than µ in the convex order. The W2-projection of µ can be easily computed when µ and ν have finite support by solving a quadratic optimization problem with linear constraints. In dimension d = 1, the projections do not depend on ρ and their quantile functions are explicit in terms of those of µ and ν. The motivation is the design of sampling techniques preserving the convex order in order to approximate Martingale Optimal Transport problems by using linear programming solvers. We prove convergence of the Wasserstein projection based sampling methods as the sample sizes tend to infinity and illustrate them by numerical experiments.
7 AYED Wided (Université Tunis El Manar) An extension of the Itô integral
Abstract: Work in progress.
BAHLALI Khaled (Université de Toulon) Unbounded BSDEs driven by z2/y and application
Abstract: We first present the existence of solutions to backward stochastic differential equations (BSDEs) 1 2 which generator has the form αt + βty + y |z| , where αt, βt, where αt and βt are suitable positive processes. This is done with a terminal value is roughly in Ł3 and is strictly positive or strictly negative. Next, we use a domination argument to derive the existence of solutions for BSDEs whose generator H is continuous and satisfies 1 2 3 0 ≤ H(t, ω, y, z) ≤ αt + βt|y| + |y| |z| . The terminal value still remains in Ł and is strictly positive or strictly negative. The uniqueness is also discussed. As a consequence, we prove the existence of a viscosity solution to 2 |∇xu| PDEs whose non-linearity is driven by u . If time permits, we discuss how the domination argument allows 1 1 2 to reduce the solvability of the BSDE with the generator H(t, ω, y, z) = y to that driven by y |z| . Joint work with Ludovic Tangpi.
BEIGLBÖCK Mathias (Universität Wien) Causal transport and its role in mathematical finance
Abstract: Causal transport plans generalize adapted processes in the same way as usual transport plans are a relaxed form of Monge-mappings. Correspondingly, one obtains an adapted / causal variant of the usual Wasserstein distance. We will review these concepts and explain why they are relevant for mathematical finance and in particular the problem of model-uncertainty.
BEN ALAYA Mohamed (Université Paris 13 Nord) Local asymptotic properties for Cox-Ingersoll-Ross process with discrete observations
Abstract: We consider a Cox-Ingersoll-Ross process whose drift coefficient depends on unknown parameters. Considering the process discretely observed at high frequency, we prove the local asymptotic normality property in the subcritical case, the local asymptotic quadraticity in the critical case, and the local asymptotic mixed normality property in the supercritical case. In this study, we require the same conditions as in the case of ergodic diffusions with globally Lipschitz coefficients studied earlier by Gobet. However, in the non-ergodic cases, additional assumptions are required.
BEN SALEM Sarah (ISFA/ Laboratoire SAF) An equilibrium model of prevention effort under distortion risk measures
Abstract: This paper studies an equilibrium model between an insurance buyer and an insurance seller, where both agents’ risk preferences are given by convex risk measures. The interaction is modeled through a Stackelberg type game, where the insurance seller plays first by offering a price, in the form of a safety loading. Then the insurance buyer chooses his optimal proportional insurance share and his optimal prevention effort in order to minimize his risk measure. The loss distribution is given by a family of stochastically ordered probability measures, indexed by the prevention effort. We show that the main conclusions of the classical results in insurance theory still hold true in this context. Finally, we solve the same problem in an adverse selection setting, where the type of insurance buyers is hidden, and is given by their risk aversion coefficient or their loss probability.
8 BENEZET Cyril (Université Paris Diderot) Partial hedging: a numerical scheme
Abstract: We introduce the partial hedging problem in mathematical finance. First, we present the stochastic optimal control problem and the PDE that the value function solves (in the viscosity sense), together with the comparison theorem, which ensures uniqueness for the solution of the PDE. We then introduce a numerical scheme to approximate the solution. This is a Piecewise Constant Policy Timestepping (PCPT). We prove its convergence and show some numerical examples. This is a joint work with Jean-François Chassagneux and Christoph Reisinger.
BLANCHARD Philippe (University of Bielefeld) ETH-approach to quantum mechanics
Abstract: This novel interpretation of QM enables us to introduce a precise notion of ""events"", to exhibit the stochastic dynamics of states, and to solve the measurement problem. This approach adds to the standard theory a simple intuitive but fundamental hypothesis.
BURZONI Matteo (ETH Zürich) Risk measures based on benchmark loss distributions
Abstract: We introduce a new class of quantile-based risk measures that generalize Value at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both the frequency and the severity of losses. Under VaR a single confidence level is assigned regardless of the size of potential losses. We allow for a range of confidence levels that depend on the loss magnitude. The key ingredient is a benchmark loss distribution (BLD), namely, a function that associate to any potential loss a maximal acceptable probability of occurrence. The corresponding risk measure determines the minimal capital injection that is required to align the loss distribution of a risky position to the target BLD. By design, one has full flexibility in the choice of the BLD profile and, therefore, in the range of relevant quantiles. Special attention is given to piecewise constant functions and to tail distributions of benchmark random losses, in which case the acceptability condition imposed by the BLD boils down to first- order stochastic dominance. We provide a comprehensive study of the main finance theoretical and statistical properties of these new risk measures with a focus on their comparison with VaR and ES. Merits and drawbacks are discussed and applications to capital adequacy, portfolio risk management and catastrophic risk are presented.
CARDALIAGUET Pierre (Université Paris Dauphine) Mean field games with a major player
Abstract: In this joint work with M. Cirant and A. Porretta, we investigate mean field games with a major player by using the point of view of the master equation. We show how to build a solution to the master equation, how the master equation gives the optimal strategies of the players and explains the mean field limit of the N-player game.
CAYÉ Thomas (Dublin City University) Utility maximization in a multidimensional market with small nonlinear price impact
Abstract: We study a portfolio choice problem in a multi-dimensional market with frictions. An investor with constant relative risk aversion invests in a market composed of a riskless asset and a multi-dimensional risky asset. Trading is hindered by sublinear price impacts that affect the asset traded but possibly also the other risky assets. In the limit for small price impact, we determine the asymptotic expansion of the value function and provide an asymptotically optimal family of trading strategies for an example. This is a work in progress with Erhan Bayraktar and Ibrahim Ekren.
9 CIPRIANO Fernanda (New University of Lisbon) On the optimal feedback controls for stochastic second grade fluids
Abstract: This article deals with a feedback optimal control problem for the stochastic second grade fluids. More precisely, we establish the existence of an optimal feedback control for the two-dimensional stochastic second grade fluids with Navier-slip boundary conditions. In addition, using the Galerkin approximations, we show that the optimal cost can be approximated by a sequence of finite dimensional optimal costs, showinwg the existence of the so-called -optimal feedback control.
COX Alexander (University of Bath) Utility maximisation with model-independent trading constraints
Abstract: In this work we consider the classical utility maximisation problem for a trader who is constrained by a model-independent portfolio constraint. Specifically, the trader aims to maximise her utility subject to the constraint that her portfolio value is bounded below when any derivative contracts are valued at their intrinsic value. Here, the intrinsic value is taken to be the model-independent super/sub-hedging price of the derivative. Using ideas of El Karoui and Meziou (2006), we are able to find explicit strategies for the trader. (Joint work with Daniel Hernandez-Hernandez).
DI NUNNO Giulia (University of Oslo) Integration with respect to Levy driven Volterra processes
Abstract: Volterra processes appear in several applications ranging from turbulence to energy finance and biological modelling. Volterra processes are in general non-semimartingales and a theory of integration with respect to such processes is in fact not standard. We present some recent results within the framework of fractional calculus and Malliavin Calculus. As illustration we consider specifically the so-called Gamma-Volterra processes. The presentation is based on joint works with: Andrea Fiacco, Erik H. Karlsen and Josep Vives.
DRAOUIL Olfa (Université Tunis El Manar) Optimal stopping of Hunt processes.
Abstract: Work in progress.
ESSAKY El Hassan (Université Cadi Ayyad) Generalized doubly reflected BSDE with predictable obstacles and stochastic quadratic growth
Abstract: This talk is concerned with the study of the existence of solutions for a one-dimensional generalized double reflected backward stochastic differential equation (GRBSDE for short) where the solution Y has to remain between two r.c.l.l. barriers L and U on [0,T ) , and its left limit has to stay respectively above and below two predictable barriers l and u on (0,T ] . The proofs of our result is based on a penalization method. This method allow us to find an equivalent form to our initial GRBSDE where its solution has to remain between two new r.c.l.l. reflecting barriers which are in fact the limit of the penalizing equations driven by the dominating conditions assumed on the coefficients. This is done without assuming any P−integrability conditions and under weaker assumptions on the input data. In particular, we construct a maximal solution for such a GRBSDE when the terminal condition ξ is only FT −measurable and the driver f is continuous with general growth with respect to the variable y and stochastic quadratic growth with respect to the variable z . It should be pointed out here that this problem is closely related to the notion of generalized Snell envelope.
10 FRIESEN Martin (Bergische University Wuppertal) On multi-type branching processes with immigration
Abstract: In this talk we adress different properties of multi-type branching processes with immigration (short CBI processes). A CBI processes is a Markov processes whose components are non-negative and their Laplace transform is an exponentially affine function of the initial state. We provide sufficient conditions that a CBI process has a transition density (w.r.t. Lebesgue measure) and, moreover, give sufficient conditions for a CBI process to be transient and for zero being a polar point. Finally we study the long-time behaviour of CBI processes, i.e. based on a suitable coupling of the trajectories we establish exponential convergence to the unique invariant measure in the Wasserstein distance.
FRITTELLI Marco (Università degli Studi di Milano) On fairness of systemic risk measures
Abstract: In our previous paper “A Unified Approach to Systemic Risk Measures via Ac- ceptance Set” we have introduced a general class of systemic risk measures that allow random allocations to individual banks before aggregation of their risks. In the present paper, we address the question of fairness of these allocations and propose a fair allocation of the total risk to individual banks. We show that the dual formulation of the minimization problem identifying the systemic risk measure provides a valuation of the random allocations, which is fair both from the point of view of the society/regulator and from the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details.
FUHRMAN Marco (Università degli studi di Milano) Optimal switching problems with an infinite set of modes: an approach by randomization and constrained BSDEs
Abstract: In classical optimal switching problems, a controller can drive the time evolution of a system choosing among a finite set of possible modes (or regimes) and switching from one mode to another at chosen random times. A set of Hamilton-Jacobi-Bellman equations (or a set of BSDEs) can be associated to this control problem, where each equation is indexed by a mode. Using a different approach introduced by Bouchard, Elie, Kharroubi and others, sometimes called randomization method, one can represent the value of the problem by a single BSDE with a constraint on the martingale part. This makes it possible to extend the BSDE representation to the case of an infinite number of modes, which is natural in many applications. This is joint work with Marie Amélie Morlais (LMM - Université du Maine - Le Mans - France).
GOZZI Fausto (Università di Roma La Sapienza) Optimal control of SPDEs and path dependent SDEs: some recent results
Abstract: We present models of optimal control of SPDEs and path dependent SDEs arising in applications. We show some recent results on them using dynamic programming techniques. Ongoing results in the Mc Kean-Vlasov case will be also briefly exposed.
HAMADÈNE Saïd (Université du Maine) Mean-field backward-forward stochastic differential equations and mean-field nonzero sum stochastic differential games.
Abstract: In this talk we discuss the problem of existence of a solution of a class of backward-forward stochastic differential equations of mean-field type. We show existence and uniqueness of a solution. As an application we discuss the problem of existence of an open-loop Nash equilibrium point for the mean-field nonzero-sum linear- quadratic stochastic differential game. Mainly, this problem turns into the resolution of a backward-forward stochastic differential equation considered in the first part for which we provide a solution. (Jww B.Djehiche, KTH, Stockholm).
11 HERNÁNDEZ SANTIBÁÑEZ Nicolás (University of Michigan) Contract theory in a VUCA world
Abstract: In this paper we investigate a Principal-Agent problem with moral hazard under Knightian uncer- tainty. We extend the seminal framework of Holmström and Milgrom by combining a Stackelberg equilibrium with a worst-case approach. We investigate a general model in the spirit of Cvitanić, Possamaï and Touzi. We show that optimal contracts depend on the output and its quadratic variation, as an extension of the works of Possamaï and Mastrolia (by dropping all the restrictive assumptions) and Sung (by considering a general class of admissible contracts). We characterize the best reaction effort of the Agent through the solution to a second order BSDE and we show that the value of the problem of the Principal is the viscosity solution of an Hamilton- Jacobi-Bellman-Isaacs equation, without needing a dynamic programming principle, by using stochastic Perron’s method.
HILBERT Astrid (Linnaeus University) A Mean-field game of pure jump type with common noise
Abstract: In this paper we study a mean field game under jump dynamics, where all the players are subject to an additional same type of Brownian noise. Moreover, We study the well-posedness and the regularity for the jump version of the stochastic partial differential equation of Mckean-Vlasov type. Finally, we show that the solution of the master equation, which is a type of second order partial differential equation in the space of probability measures, provides an approximate Nash-equilibrium. Joint work with Ch. Grün and R. Basna.
HILLAIRET Caroline (ENSAE) Construction of an aggregate consistent utility.
Abstract: Our aim is to describe globally the behavior and preferences of heterogeneous agents. The starting point is the aggregate wealth of a given economy, with a given repartition of the wealth among investors, which is not necessarily Pareto optimal. We propose a construction of an aggregate forward utility, market consistent, that aggregates the marginal utility of the heterogeneous agents. This construction is based on the aggregation of the pricing kernels of each investor. As an application we analyze the impact of the heterogeneity and of the wealth market on the yield curve.
HORVATH Blanka (University College London) Learning rough volatility
Abstract: Calibration time being the bottleneck for models with rough volatility, we present ways for sub- stantial speed-ups, along every step of the calibration process: In a rst step we describe a powerful numerical scheme (based on functional central limit theorems) for pricing a large family of rough volatility models. In a second step we discuss various machine learning methods that signi cantly reduce calibration time for these models. By simultaneously calibrating several (classical and rough) models to market data, we re-confirm as a byproduct of our calibration results, that volatility is rough, calibration performance being best for very small Hurst parameters in a multitude of market scenarios.
HU Kaitong (École Polytechnique) Continuous-time Principal-Agent Problem in a Linear Partially Observed System
Abstract: In this presentation, we are going to consider the Principal-Agent problem in a linear partially observed system, namely, a linear output process one part of which is non-observable neither by Principal nor the Agent. Firstly, we shall demonstrate that the solvability of the Agent’s optimization problem can be reduced to the solvability of the system of strongly coupled FBSDEs using variational calculus in weak formulation and then a corresponding sufficient condition of the problem will be given as well. At last, we will compute the optimal contract for the Principal’s problem.
12 HUBERT Emma (Université Paris-Est Marne-la-Vallée) Optimal remuneration of correlated consumers in electricity demand Abstract: We formulate the problem of demand response contracts in electricity markets as a Principal-Agent problem with moral hazard. The Principal is a risk–averse or risk-neutral producer subject to energy generation cost and to consumption volatility cost due to limited flexibility of production. The Principal observes in continuous-time the consumption of risk–averse Agents who are numerous and considered to be interacting in a mean-field game, but she does not observe the efforts they make to reduce their consumption and the volatility of their consumption in their different usages. We explore whether allowing for contracts that can be indexed on both the deviation of a given consumer and aggregate statistics of the consumptions of other similar consumers can be helpful in practice for the Principal.
KAZI TANI Nabil (ISFA) An open problem in ruin theory and its diffusion approximation regime Abstract: The De Vylder and Goovaerts conjecture is an open problem in risk theory, stating that the finite time ruin probability in a standard risk model is greater or equal to the corresponding ruin probability evaluated in an associated model with equalized claim amounts. Equalized means here that the jump sizes of the associated model are equal to the average jump in the initial model between 0 and a terminal time T . In this talk, we consider the diffusion approximations of both the standard risk model and the associated risk model. We prove that the associated model, when conveniently renormalized, converges in distribution to a gaussian process satisfying a simple SDE. We compute the probability that this diffusion hits the level 0 before time T and compare it with the same probability for the diffusion approximation for the standard risk model. Then we conclude that the De Vylder and Goovaerts conjecture holds true for these diffusion limits. This is a joint work with Stefan Ankirchner (University of Jena) and Christophette Blanchet-Scalliet (Ecole Centrale de Lyon and ICJ).
KREMER Jonas (Bergische University Wuppertal) Existence of limiting distribution for affine processes Abstract: In this talk, sufficient conditions are discussed for the existence of limiting distribution of a conserva- m n tive affine process on the canonical state space × , where m, n ∈ 0 with m + n > 0. Our main theorem R>0 R Z> extends and unifies some known results for OU-type processes on Rn and one-dimensional CBI processes (with state space R>0). To prove our result, we combine analytical and probabilistic techniques; in particular, the stability theory for ODEs plays an important role. The talk is based on a joint work with Peng Jin and Barbara R¨"udiger.
KUPPER Michael (Universität Konstanz) Computation of optimal transport and robust risk aggregation with neural networks Abstract: We present a widely applicable approach to solving (multi-marginal, martingale) optimal transport and related problems via neural networks. The core idea is to penalize the optimization problem in its dual for- mulation and reduce it to a finite dimensional one which corresponds to optimizing a neural network with smooth objective function. We present numerical examples from optimal transport, and bounds on the distribution of a sum of dependent random variables. As an application we focus on the problem of risk aggregation under model uncertainty. The talk is based on joint work with Stephan Eckstein and Mathias Pohl.
MAGGIS Marco (Università degli Studi di Milano) Stochastic dynamic utilities and inter-temporal preferences Abstract: We propose an axiomatic approach which economically underpins the representation of dynamic preferences in terms of a stochastic utility function, sensitive to the information available to the decision maker. Our construction is recursive and based on inter-temporal preference relations, whose characterization is inpired by the original intuition given by Debreu’s State Dependent Utilities (1960).
13 MARTIN Jessica (Université de Toulouse) A revisit of the Borch rule for the Principal-Agent Risk-Sharing problem
Abstract: In this talk we provide a new approach to tackle the Principal-Agent Risk-Sharing problem using optimal stochastic control technics. Our analysis relies on an optimal decomposition of the expected utility of the Principal in terms of the reservation utility of the Agent. In particular, this allows us to derive the Borch rule as a necessary optimality condition for this decomposition to hold, which sheds a new light on this economic concept. As a by-product, this approach provides a class of risk-sharing plans that satisfy the Borch rule; class to which the optimal plan belongs.
MEZERDI Brahim (University of Biskra) On stochastic control of mean-field systems
Abstract: In this talk, we deal with optimal control of systems driven by mean-field stochastic differential equations. These equations are obtained as limits of interacting particle systems, as the number of particle tends to infinity. This mean-field equation, represents in some sense the average behavior of the infinite number of particles. Since the earlier papers by Lasry-Lions and Huang-Malhamé-Caines, mean-field control theory and mean-field game theory has raised a lot of interest, motivated by applications to various fields such as game theory, mathematical finance, communications networks, management of oil ressources. Mean-field control problems occur in many applications, such as in a continuous-time Markowitz’s mean–variance portfolio selection model where the variance term involves a quadratic function of the expectation. We are interested in relaxed controls which are measure valued processes. We prove that the strict and relaxed control problems have the same value function and that an optimal relaxed control exists. Moreover, we establish necessary conditions for optimality in the form of a relaxed stochastic maximum principle, obtained via the first and second order adjoint processes.
MRA’D Mohamed (Université Paris 13 Nord) Markovian dynamic utility from monotonic characteristic
Abstract: Given a stochastic process X , the aim of this work is to establish a necessary and sufficient condition for the existence of a Markovian utility function, whose optimal process is the data X .
OBATA Nobuaki (Tohoku University) Component sizes in random graphs
Abstract: We report some statistics of component sizes in configuration models with power law degree distri- butions by means of numerical simulation. This is a joint work with Takehisa Hasegawa (Ibaraki University, Japan).
OBŁÓJ Jan (University of Oxford) On numerical techniques for robust methods of pricing and hedging
Abstract: I discuss recent works related to the effort of creating numerical techniques for robust approach to pricing and hedging. In the robust approach both time series data as well as market prices for options are taken into the account. This is novel in comparison to the classical approach where these two types of data are usually considered under different probability measures (real world vs risk neutral). Using time-series data, I consider the toy problem of using historical returns to estimate the superhedging price of an option in a one-step model. We introduce several estimators and discuss their consistency and robustness, both statistical as well as financial. We discuss in detail the natural plug-in estimator, show it is consistent and estimate its rates of convergence, but also show that it is not robust. To address this, we propose estimators obtained using Wasserstein balls which turn out to be (in a suitable sense) statistically robust and also, under some regularity of option’s payoff, financially robust. This is a joint work with Johannes Wiesel. Using option prices data leads to the so-called
14 martingale optimal transport (MOT) problem. We prove that the MOT problem can be approximated through a sequence of linear programming (LP) problems which result from a discretisation of the marginal distributions combined with a suitable relaxation of the martingale constraint. Specialising to the one-step model in dimension one, we provide an estimation of the convergence rate and detailed computational algorithms. This is a joint work with Gaoyue Guo. At the end I present outlook for merging different numerical techniques into a single comprehensive approach. Non linear optimal stopping problem and Re ected BSDE in the predictable setting
OUKNINE Youssef (University Mohammed 6 Polytechnique, Marrakech, Morocco) Non linear optimal stopping problem and Reflected BSDE in the predictable setting
Abstract: In the first part of this paper, we study RBSDEs in the case where the filtration is non quasi-left continuous and the lower obstacle is given by a predictable process. We prove the existence and uniqueness by using some results of optimal stopping theory in the predictable setting, some tools of general theory of processes as the Mertens decomposition of predictable strong supermartingale. In the second part we introduce an optimal stopping problem indexed by predictable stopping times with non linear predictable g−expectation induced by an appropriate BSDE. We establish some useful properties of Ep,g−supremartingales. Moreover, we characterize the predictable value function in terms of the first component of RBSDEs studied in the first part.
ØKSENDAL Bernt (University of Oslo) Optimal control of SPDEs with space-mean dynamics
Abstract: We study systems with dynamics described by stochastic partial differential equations (SPDEs) combined with space-mean (""nearest neighbours"") terms. Such systems are interesting as models for population growth and epidemiology, for example. We give a sufficient and a necessary maximum principle for the optimal control of such systems. The results are illustrated by examples. This is joint work with Nacira Agram (Biskra University, Algeria) and Astrid Hilbert (LNU Växjö, Sweden).
PETTERSSON Roger (Linnaeus University) On an epidemic diffusion model
Abstract: We consider an epidemic model describing the fraction of infected individuals driven by a Wiener process, as suggested in Iacus (2008). The model involves a transmission rate, a recovery rate, a transmission rate from an external source, and a population density parameter respectively. Depending on the different choices of the parameters, different properties of a solution will be obtained, in particular asymptotics. We also compare with a diffusion approximation of a related discrete-valued Markov process in continuous time.
PONTIER Monique (Université Paul Sabatier) PDE for joint law of the pair of a continuous diffusion and its running maximum
Abstract: Let X be a d-dimensional diffusion process and M the running supremum of the first component. In this paper, we first show that for any t > 0, the law of the pair (Mt,Xt) admits a density with respect to Lebesgue measure, which can be characterized as a weak solution of a partial differential equation on an open set. In the one-dimensional case, this density is a weak solution of a PDE on the set {(m, x) ∈ R2, m ≥ sup(x, 0)} with specified boundary conditions.
PRÖMEL David (University of Oxford) On Skorokhod embeddings and Poisson equations
15 Abstract: The Skorokhod embedding problem for a given stochastic process and a target probability measure ask to find a stopping time such that stopped process is distributed as the given target probability measure. While this problem is well-studied for continuous processes like Brownian motion, there are only a few known results in the case stochastic processes with jumps. For general Levy processes, we propose necessary and sufficient conditions for the existence of a solution to the Skorokhod embedding problem in terms of the Poisson equation associated to the adjoint operator of the Levy process. Our approach provides an explicit construction of the stopping time and is based on new uniqueness results for Fokker-Planck equations with degenerates coefficients. The talk is based on a joint work with Leif Döring, Lukas Gonon and Oleg Reichmann.
RÜDIGER Barbara (Bergische University Wuppertal) The Enskog equation and the Enskog-Boltzmann Process
Abstract: In a previous article we derived a McKean-Vlasov equation for which the solution is distributedaccord- ing to the Boltzmann (-Enskog) equation. We call its solution the Boltzmann (-Enskog) process. Under suitable conditions the Existence of the Boltzmann (-Enskog) process is proven for all cases including hard spheres, soft and hard potentials. Uniqueness of the time marginals of the Enskog process is proven under suitable moment conditions. The results are obtained in two joint works with S. Albeverio, P. Sundar, and two joint work with M. Friesen and P. Sundar.
SALHI Naoufel (Université Tunis El Manar) Some facts about Gaussian integrators
Abstract: We recall the definition of Gaussian integrators and give some examples and some properties. We precise their relation with Skorokhod integral then we show some facts about their local times and self-intersection local times.
SCOTTI Simone (Université Paris Diderot) An optimal dividend control problem with investment opportunities and business cycle
Abstract: This paper concerns with the problem of determining an optimal control on the dividend under debt constraints and investment opportunities in an economy with business cycles. We allow the company to accept or reject investment/desinvestment opportunities arriving at random times with random sizes by increasing/decreasing its outstanding indebtedness, which would impact its capital structure and risk profile. Similarly, we formulate this problem as a bi-dimensional singular control problem under regime switching in presence of jumps. We obtain an explicit condition in order to have that the value function is finite. We use a viscosity solution approach to get qualitative descriptions of the solution. We further enrich our studies with a Picard scheme converging to the value function and having explicit solution. We also provide some numerical illustrations.
SGARRA Carlo (Politecnico Milano) A forward price model for power markets based on branching processes
Abstract: We propose and investigate a market model for forward prices in power markets, including most basic features exhibited by previous models and taking into account self-exciting properties. The model proposed extends Hawkes-type models by introducing a two-fold integral representation property.A Random Field approach was already exploited by Barndorff-Nielsen, Benth and Veraart who adopted an Ambit Field framework for describing the forward dynamics of power prices. The novelty contained in our approach consists in combining the basic features of both Branching Processes and Random Fields in order to get a realistic and parsimonious model setting. We discuss the no-arbitrage issue of the present modelling framework. We outline a possible methodology for parameters estimation. We illustrate by graphical representation the main achievements of this approach. Joint work by C. Callegaro„ A. Mazzoran, S. Scotti and C. Sgarra.
16 SILVA José Luis (University of Madeira) The Stein Characterization of M-Wright Distributions
Abstract: In this talk we use the Stein method to characterize the M−Wright distribution M1/3 and its symmetrization. The associated Stein operator corresponds to the general Airy equation and the Stein equation is nothing but a general inhomogeneous Airy equation.
STREIT Ludwig (Bielefeld University) Fractional Brownian loops and trees
Abstract: Recent results on their construction and properties.
TAN Xiaolu (Université Paris Dauphine) On the planification problem in Mean Field Games
Abstract: In the context of a generalized version of mean field games, with possible control of the diffusion coefficient, we consider the planification problem introduced by P.L. Lions: given a pair (µ, ν) of starting and target probability measures on the state space, find a specification of the game problem which induces ν at the mean field game equilibrium. We provide conditions on (µ, ν) which guarantee existence of a planification plan. We next introduce an optimal planification problem which allows to select some remarkable planification plan. Our main result reduces this optimal planification problem into an optimal transport problem along a McKean-Vlasov controlled SDE.
TANKOV Peter (ENSAE) Mean field games of optimal stopping: a relaxed control approach
Abstract: We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem where the reward function depends on the density of states of agents still present in the game. Placing ourselves in the framework of relaxed optimal stopping, we prove the existence and uniqueness of the mixed Nash equilibrium. Further, we provide a criterion under which the optimal strategies are pure strategies, and present a numerical method for computing the equilibrium. Applications to mathematical finance and financial economics will be briefly discussed.
VON WALDENFELS Wilhem (Heidelberg University) The H−Theorem in radiation transfer
Abstract: The radiation transfer equation describes the random walk of photons through a gas. It defines an entropy transport. The divergence of the entropy current is positive. This is an analogy of Boltzmann’s H-theorem.
ZERVOS Mihail (London School of Economics) Self-enforcing risk-sharing arrangements in a continuous-time endowment economy
Abstract: We characterise efficient risk-sharing under two-sided limited commitment in a continuous-time endowment economy. We take a dual approach to the problem based on the Kuhn-Tucker multipliers on the participation constraints, and establish a strong duality result in a general setting. In an application, agents have the same time-separable preferences, aggregate endowment is constant, and endowment shares are driven by a mean-reverting diffusion process. The value function for the dual problem is homogenous and the relevant co-state variable is a ‘modified’ relative Pareto-Negishi weight, which determines the consumption allocation. We analyse the HJB equation associated with the singular control problem and solve for the free boundaries that determine regions of the state space where the participation constraints are binding. In particular, we show that, for some parameter values, perfect risksharing can be sustained despite limited commitment.
17 List of participants
Acciaio, Beatrice. London School of Economics, [email protected]. Agram, Nacira. University of Biskra, [email protected]. Aïd, René. Université Paris Dauphine, [email protected]. Alfonsi, Aurélien. École des Ponts ParisTech, [email protected]. Amami, Rim. Université Tunis El Manar, [email protected]. Ayed, Wided. Université Tunis El Manar, [email protected]. Bahlali, Khaled. Université de Toulon, [email protected]. Barrasso, Adrien. École Polytechnique, [email protected]. Beiglböck, Mathias. Universität Wien, [email protected]. Ben Alaya, Mohamed. Université Paris 13 Nord, [email protected]. Ben Salem, Sarah. ISFA/ Laboratoire SAF, [email protected]. Benezet, Cyril. Université Paris Diderot, [email protected]. Blanchard, Philippe. University of Bielefeld, [email protected]. Bouchard, Bruno. Université Paris Dauphine, [email protected]. Boukhris, Radhouane. Université Tunis El Manar, [email protected]. Burzoni, Matteo. ETH Zürich, [email protected]. Cardaliaguet, Pierre. Université Paris Dauphine, [email protected]. Cayé, Thomas. Dublin City University, [email protected]. Chaari, Sonia. University of Carthage, [email protected]. Chevalier, Etienne. ENSIEE, [email protected]. Cipriano, Fernanda. New University of Lisbon, [email protected]. Claisse, Julien. Université Paris Dauphine, [email protected]. Cox, Alexander. University of Bath, [email protected]. Di Nunno, Giulia. University of Oslo, [email protected]. Djete, Fabrice. Université Paris Dauphine, [email protected]. Doldi, Alessandro. Università degli Studi di Milano, [email protected]. Draouil, Olfa. Université Tunis El Manar, [email protected]. Ekeland, Ivar. Université Paris Dauphine, [email protected]. El Karoui, Nicole. Université Pierre et Marie Curie, [email protected]. Essaky, El Hassan. Université Cadi Ayyad, [email protected]. Ettaieb, Aymen. Université de Kairouan, [email protected]. Fakhouri, Imade. Ecole Mohamed 6 Marrakech, [email protected]. Farhat, Heythem. Ecole Polytechnique, [email protected]. Friesen, Martin. Bergische University Wuppertal, [email protected]. Frittelli, Marco. Università degli Studi di Milano, [email protected]. Fuhrman, Marco. Università degli studi di Milano, [email protected]. Gozzi, Fausto. Università di Roma La Sapienza, [email protected]. Hachaichi, Rached. Université Tunis El Manar, [email protected]. Hamadène, Saïd. Université du Maine, [email protected]. Haouala, Ezzeddine. Université Tunis El Manar, [email protected]. Hernández Santibáñez, Nicolás. University of Michigan, [email protected]. Hilbert, Astrid. Linnaeus University, [email protected]. Hillairet, Caroline. ENSAE, [email protected]. Horvath, Blanka. University College London, [email protected]. Hu, Kaitong. Ecole Polytechnique, [email protected]. Hubert, Emma. Université Paris-Est Marne-la-Vallée, [email protected]. Jeanblanc, Monique. Université d’Évry-Val-d’Essonne, [email protected]. Jendoubi, Souheyl. Carthage University, [email protected]. Kazi Tani, Nabil. ISFA, [email protected]. Kehl, David. Bergische University Wuppertal, . Kharroubi, Idris. Université Pierre et Marie Curie, [email protected]. Khedher, Asma. University of Amsterdam, [email protected]. Kremer, Jonas. Bergische University Wuppertal, [email protected].
18 Kupper, Michael. Universität Konstanz, [email protected]. Maggis, Marco. Università degli Studi di Milano, [email protected]. Martin, Jessica. Université de Toulouse, Jessica Martin
Organizing Committee
Heythem Farhat (Ecole Polytechnique), Habib Ouerdiane (University of Tunis El Manar), José Luís da Silva (University of Madeira).
Conference Organizers
Dylan Possamaï (Columbia University, USA) Email: [email protected] Bernt Øksendal (University of Oslo, Norway) E-mail: [email protected] Habib Ouerdiane (University of Tunis El manar, Tunisia) Email: [email protected] Mer Soner (ETH Zürich), Email: [email protected] Nizar Touzi (Ecole Polytechnique de Paris, France) Email: [email protected]
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